Bear in mind that oil in this context is generally slow-burning lamp oil, not any kind of modern refined petroleum product. From a game balance perspective, a flask available for one silver piece shouldn’t compete in power with magic spells that do fire damage.
Here are the rules for oil flasks from the Player’s Handbook:
Oil usually comes in a clay flask that holds 1 pint. As an action, you can splash the oil in this flask onto a creature within 5 feet of you or throw it up to 20 feet, shattering it on impact. Make a ranged attack against a target creature or object, treating the oil as an improvised weapon. On a hit, the target is covered in oil. If the target takes any fire damage before the oil dries (after 1 minute), the target takes an additional 5 fire damage from the burning oil. You can also pour a flask of oil on the ground to cover a 5-foot-square area, provided that the surface is level. If lit, the oil burns for 2 rounds and deals 5 fire damage to any creature that enters the area or ends its turn in the area. A creature can take this damage only once per turn.
The last, bolded sentence is the one we really care about, but the rest of the rules are important for context. Of course, since there’s some ambiguity in this critical sentence, it’s up to each individual DM to determine how they will resolve that ambiguity in their own game. What follows is my reasoning about how to interpret this language and how I would rule in my own games.
First, it should be noted that covering a creature with oil from one flask and setting them on fire only does 5 fire damage.
Second, if a creature stands in a square of ignited oil for their entire turn, they also take only 5 fire damage.
We can infer from these two points that in general, being exposed to burning oil for a whole turn will do 5 fire damage.
Let’s then consider a slightly more involved scenario: a doorway whose space is filled with ignited oil. If a character’s turn consists of running through the doorway, grabbing an object on the floor, and then running back out, it’s clear from the rules that the character will take only 5 points of fire damage, since “A creature can take this damage only once per turn”.
This is also consistent with the idea that a single turn’s exposure to burning oil will do 5 fire damage.
In all of the above cases, the scope of “this damage” is clearly confined to meaning “the damage caused by one specific flask of oil”. But it’s not clear that that’s the limit of what “this damage” might mean. “This damage” could be taken to mean “damage from all burning oil sources”.
Let’s revisit our burning doorway scenario, but have the character exit the room through a second doorway, also filled with ignited oil. How much damage should the character take?
An argument can be made for either 5 fire damage or 10 fire damage. In the latter case, “this damage” is interpreted to mean “fire damage from this specific flask of oil”.
Given that there’s no other language present to disambiguate “this damage”, we can see which interpretation is more consistent with other information we have about how burning oil is intended to behave.
We know that the cases where a creature has the maximum amount of time possible exposed to burning oil in a turn results in 5 fire damage. These are the first two examples considered above, where a creature is either doused in oil and ignited, or where they stand immobile in burning oil for a whole turn.
So what interpretation of the meaning of “a creature can take this damage only once per turn” is most consistent with the other rules that limit whole-turn damage from burning oil to 5 points? To my mind, it seems most consistent to cap the damage to a creature from all burning oil sources to 5 fire damage per turn.
In the other interpretation, where fire damage is limited per oil source, characters are incentivized to act in ways that don’t make much sense in context. For example, if a whole corridor is full of burning oil, a character will take less damage if they stand immobile in one square than if they sprint across several squares.
Finally, given the relative ease that characters can obtain burnable oil, limiting the total per-creature per-turn damage seems more in keeping with the intended power level.